Puzzle of Regge Dips

The formalism giving rise to Regge dips is reviewed, and the modifications due to singular residues are discussed. The $\pi N$ charge-exchange reaction is parametrized to allow for a singular $\rho$ residue as ${\alpha }_{\rho }$ passes through zero, but the fits show that the singularity is very weak. A model of a singular residue is presented, and it is argued that perturbation-theory models are at variance with the experimentally observed behavior of the $\rho$ residue.